📚 The Modulus Function | 模函数（绝对值函数）
The modulus function, also called the absolute value function, is one of the core topics in the Edexcel A-Level Mathematics specification. It appears in algebra, graph sketching, equation solving and inequality problems, and it frequently combines with quadratics, coordinate geometry and trigonometry. Building a solid understanding of |x| as the distance from zero allows students to handle more demanding expressions such as |f(x)| and f(|x|), and it helps them avoid the classic errors that quietly cost marks in the exam. This article explains the definition, the piecewise form, the key algebraic properties, graph transformations, solving techniques for equations and inequalities, and the most common exam pitfalls, with worked examples throughout.
模函数（也称绝对值函数）是爱德思 A-Level 数学大纲中的核心主题之一。它会出现在代数、图像绘制、解方程和不等式问题中，并常与二次函数、坐标几何和三角函数结合。把 |x| 扎实地理解为 到零的距离，能帮助学生处理更复杂的表达式，如 |f(x)| 与 f(|x|)，并避免那些悄悄吃掉分数的常见错误。本文讲解定义、分段形式、核心代数性质、图像变换、方程与不等式的求解技巧，以及最常见的考试陷阱，并配有例题讲解。
1. Definition and Notation | 定义与符号
For any real number x, the modulus (or absolute value) of x is written as |x|. Geometrically, |x| is the distance from the point x to 0 on the real number line, and since distance is never negative, |x| is never negative either. More generally, |a − b| is the distance between the two points a and b on the number line, which is why modulus expressions appear so often in distance and interval problems.
对于任意实数 x，x 的模（或称绝对值）记作 |x|。从几何上看，|x| 是数轴上点 x 到 0 的距离；由于距离永远不为负，|x| 也永远不为负。更一般地，|a − b| 表示数轴上两点 a 与 b 之间的距离，这正是模表达式在距离与区间问题中频繁出现的原因。
|x| = x for x ≥ 0, |x| = −x for x < 0
Algebraically, the definition splits into two cases: if x is zero or positive, |x| equals x itself; if x is negative, |x| als the opposite of x, which turns the negative input into a positive output. For example, |5| = 5, |−3| = 3, |0| = 0 and |−0.25| = 0.25.
从代数上看，定义分为两种情况：如果 x 为零或正数，|x| 等于 x 本身；如果 x 为负数，|x| 等于 x 的相反数，从而把负的输入变成正的输出。例如，|5| = 5，|−3| = 3，|0| = 0，|−0.25| = 0.25。
One small but important exam point is notation. The bars in |x| are modulus bars, not brackets and not the determinant symbol used for matrices. When a question asks for the modulus of a whole expression, such as |3x − 7|, the bars must enclose everything that is being made positive.
一个很小但很重要的考试要点是记号。|x| 中的竖线是模的记号，既不是普通括号，也不是矩阵中的行列式符号。当题目要求整个表达式的模时，例如 |3x − 7|，竖线必须把需要变为正数的全部内容都括进去。
2. The Modulus as a Piecewise Function | 分段函数表示
odulus function is the simplest example of a piecewise function in the A-Level course. Written in full, it has two branches joined at the origin, and both branches must be stated for full marks in a definition question.
由于规则在 x = 0 处发生改变，模函数是 A-Level 课程中最简单的分段函数例子。完整写出时，它由两支组成并在原点相接；在定义类问题中，两支都必须写出才能得满分。
f(x) = |x| = x for x ≥ 0 ; f(x) = |x| = −x for x < 0
The domain of f(x) = |x| ever be negative. The piecewise idea extends directly to more complicated expressions. For |2x − 6| 6 = 0, that is at x = 3, so the two branches are 2x − 6 for x ≥ 3 and 6 − 2x for x < 3.
f(x) = |x| 的定义域是全体实数，值域是 [0, ∞)，因为输出永远不可能为负。分段的思路可以直接推广到更复杂的表达式。对于 |2x − 6|，竖线内的表达式在 2x − 6 = 0 即 x = 3 处变号，因此两支分别是：当 x ≥ 3 时为 2x − 6，当 x < 3 时为 6 − 2x。
The same reasoning gives the very useful translation form |x − a| n x < a. This is exactly the distance from x to the point a, and recognising it saves a great deal of time in both equation and inequality questions.
同样的推理给出非常有用的平移形式 |x − a|：当 x ≥ a 时等于 x − a，当 x < a 时等于 a − x。它恰好表示 x 到点 a 的距离，识别出这一点在方程与不等式题目中能节省大量时间。
3. Key Properties of the Modulus | 模的核心性质
These algebraic properties are used constantly in later questions, so they are worth memorising as a block. Each one follows directly from the definition, and each one can be justified quickly in an exam if a proof is requested.
这些代数性质在后续题目中被反复使用，因此值得整块记忆。每一条都由定义直接推出，若考试要求证明，也可以快速说明理由。
- |x| ≥ 0 for every real x, and |x| = 0 only when x = 0.
对任意实数 x 都有 |x| ≥ 0，且仅当 x = 0 时 |x| = 0。 - |−x| = |x|, so changing the sign of the input never changes the modulus.
|−x| = |x|，因此输入取相反数不会改变模。 - |xy| = |x| × |y|, the modulus distributes over multiplication.
|xy| = |x| × |y|，模对乘法可以分配。 - |x ÷ y| = |x| ÷ |y| for y ≠ 0.
当 y ≠ 0 时，|x ÷ y| = |x| ÷ |y|。 - |x|² = x² and |x| = √x², so squaring removes the modulus.
|x|² = x² 且 |x| = √x²，因此平方可以去掉模。 - |x + y| ≤ |x| + |y|, the triangle inequality.
|x + y| ≤ |x| + |y|，即三角不等式。 - |x − y| ≥ ||x| − |y||, the reverse triangle inequality.
|x − y| ≥ ||x| − |y||，即反向三角不等式。
A warning is essential here: |x + y| is generally NOT equal to |x| + |y|. Taking x = 3 and y = −5 gives |x + y| = |−2| = 2 but |x| + |y| 5 = 8. Students who assume additivity lose marks on both algebra and inequality questions.
这里必须提醒：|x + y| 一般并不等于 |x| + |y|。取 x = 3、y = −5，则 |x + y| = |−2| = 2，而 |x| + |y| = 3 + 5 = 8。假设模可以直接相加的学生会在代数题和不等式题上双重失分。
4. The Graph of y = |x| | y = |x| 的图像
The graph of y = |x| t passes through (1, 1), (2, 2), (−1, 1) and (−2, 2), and it is symmetric about the y-axis, that is, it is an even function.
y = |x| 的图像是一个 V 形，其顶点（即最低点）在原点 (0, 0)。两条臂与 x 轴成 45° 上升，整条曲线都位于 x 轴之上或与 x 轴重合。它经过 (1, 1)、(2, 2)、(−1, 1)、(−2, 2)，并且关于 y 轴对称，也就是说它是一个偶函数。
ves at all. This single rule, keep the positive part and fold the negative part upward, generates every graph of the form y = |f(x)|.
绘制这条曲线最快的方法是先画 y = x，然后把位于 x 轴下方的部分关于 x 轴翻折上去。原本就在 x 轴上方的部分完全不动。这条简单的规则——保留正值部分、把负值部分向上翻折——可以生成所有 y = |f(x)| 形式的图像。
Because the gradient jumps from −1 to +1 at the origin, y = |x| use no trouble, but any question involving the gradient of the modulus must exclude x = 0.
由于斜率在原点处从 −1 突跳为 +1，y = |x| 在 x = 0 处有一个尖角，因而在该点不可导。这条曲线处处连续，因此极限与连续性问题不会带来麻烦，但任何涉及模函数导数的问题都必须排除 x = 0。
5. Graphs of y = |f(x)| | y = |f(x)| 的图像
To sketch y = |f(x)| int where f(x) ≥ 0 unchanged, and reflect every point where f(x) < 0 in the x-axis. The
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